integral proofing

Oct 2012
41
0
uk
let f(x,t)=xe^(-xt).show that the integral I(x)=∫f(x,t)dt (integration from 0 to infinite)exists for all x>=0 . is x->I(x) continuous on [0,infinite)
what should i use here to prove the integral exist ???can someone give me the detail expanlation???
 

Plato

MHF Helper
Aug 2006
22,507
8,664
let f(x,t)=xe^(-xt).show that the integral I(x)=∫f(x,t)dt (integration from 0 to infinite)exists for all x>=0 . is x->I(x) continuous on [0,infinite) what should i use here to prove the integral exist ???can someone give me the detail expanlation???
\(\displaystyle \int_0^\infty {xe^{ - xt} dt} = \lim _{b \to \infty } \left( {\left. { - e^{ - xt} } \right|_{t = 0}^{t = b} } \right)=\lim _{b \to \infty } \left( {1 - e^{-xb} } \right) = ?\)
 
Oct 2012
41
0
uk
thank you,then the limit is 1 so the integral exists,but how about the continuous part, for now x->I(x) ,I(X)=1 it seems ,it is continuous?? i am not sure
 

Plato

MHF Helper
Aug 2006
22,507
8,664
thank you,then the limit is 1 so the integral exists,but how about the continuous part, for now x->I(x) ,I(X)=1 it seems ,it is continuous?? i am not sure
Frankly, I have the same question about \(\displaystyle I(x)\).
We know that \(\displaystyle I(x)\ge 0\). But I have no idea how it fits into the question.
 
Mar 2010
1,055
290
Isn't \(\displaystyle I(x)=1\) for \(\displaystyle x>0\) and \(\displaystyle I(x)=0\) for \(\displaystyle x=0\)? So it would be continuous on \(\displaystyle (0,\infty)\) and discontinuous at 0, right?

- Hollywood